Lemma 4.3 (cf. [1]}) 
For every \(\delta >0\)  and every \(q\in \mathbb {N}\) , there exists
\(C_0=C_0(q,\delta )>0\)  such
that for all \(t\ge 1\)
\(\mathbb {P}\Big (\sup _{\begin{array}{c}C_0\ln \lfloor t\rfloor \le |h|\le \lceil t\rceil \cdot \varepsilon ^*(t),\\ v\in V\end{array}}\Big | L(\overline{\eta }(v))- \frac{1}{h}\sum _{i=vt+1}^{ vt+h}L_i^\zeta (\eta _{v t}^{\zeta }(v))\Big | >\delta , \big (vt\ge {constHn}\ \forall v\in V \big ) \Big ) \le c t^{-q}.\) 

